Single Imputation Methods and Confidence Intervals for the Gini Index

This research has been partially supported by the Ministry of Economy, Industry and Competitiveness, the Spanish State Research Agency (SRA) and European Regional Development Fund (ERDF) (project reference ECO2017-86822-R). This research has been partially supported by the Ministry of Economy, Industry and Competitiveness, the Spanish State Research Agency (SRA) and European Regional Development Fund (ERDF) (project reference ECO2017-84138-P).The problem of missing data is a common feature in any study, and a single imputation method is often applied to deal with this problem. The first contribution of this paper is to analyse the empirical performance of some traditional single imputation methods when they are applied to the estimation of the Gini index, a popular measure of inequality used in many studies. Various methods for constructing confidence intervals for the Gini index are also empirically evaluated. We consider several empirical measures to analyse the performance of estimators and confidence intervals, allowing us to quantify the magnitude of the non-response bias problem. We find extremely large biases under certain non-response mechanisms, and this problem gets noticeably worse as the proportion of missing data increases. For a large correlation coefficient between the target and auxiliary variables, the regression imputation method may notably mitigate this bias problem, yielding appropriate mean square errors. We also find that confidence intervals have poor coverage rates when the probability of data being missing is not uniform, and that the regression imputation method substantially improves the handling of this problem as the correlation coefficient increases.Ministry of Economy, Industry and Competitiveness Spanish State Research Agency (SRA)European Commission ECO2017-84138-

Indeks gini derajat pada graf unit dari ring bilangan bulat modulo

INDONESIA: Misalkan R suatu ring dengan unsur kesatuan, e∈R suatu unsur kesatuan, dan U(R) adalah suatu himpunan anggota unit dari R (U(R) = {x∈R|x∙y=e,y∈R}). Graf unit G(R) adalah graf dengan semua elemen R sebagai titik dan dua titik yang berbeda, x dan y, terhubung langsung jika dan hanya jika x+y∈U(R). Indeks gini derajat pada graf H merupakan jumlah dari selisih dua derajat titik sebarang pada graf H yang dibagi dengan perkalian antara jumlah titik dan rata-rata derajat titik pada graf H atau: GD(H)=∑_(u,v∈V(H)@u≠v)▒|deg(u)-deg(v)|/((n^2)E[D^* (H)]) dengan deg(u) dan deg(v) masing-masing adalah derajat titik u dan v pada graf H, n adalah kardinalitas dari himpunan titik di H, dan E[D^* (H)] adalah rata-rata derajat titik pada graf H. Penelitian ini bertujuan untuk menentukan formula indeks gini derajat pada graf unit dari ring bilangan bulat modulo 3p dengan p bilangan prima dengan cara menentukan unit pada ring bilangan bulat modulo 3p dan derajat titik pada graf unit dari ring bilangan bulat modulo 3p. Hasil penelitian ini adalah sebagai berikut: GD(G(Z_3p))=█(0;p=2@ 1/24;p=3@ (p+2)/3p(3p-1);p>3) ENGLISH: Let R be a ring with unity, e∈R be a unity, and U(R) is a set of units in R (U(R) = {x∈R|x∙y=e,y∈R}). The unit graph G(R) is a graph with all of elements in R as the vertices and two distinct vertices, x and y, are adjacent if and only if x+y∈U(R). The degree gini index of a graph H is the sum of the difference of any two degrees on a graph H which divided by the multiple of the number and the mean of the degrees on a graph H or: GD(H)=∑_(u,v∈V(H)@u≠v)▒|deg(u) deg(v)|/((n^2)E[D^* (H)]) where deg(u) and deg(v) is the degree of u and v on a graph H respectively, n is the cardinality of the set of vertices on a graph H, and E[D^* (H)] is a mean of the degrees on a graph H. This research aims to determine the formula of the degree gini index on the unit graph of the integer ring modulo 3p with p is a prime number by determining the units in the integer ring modulo 3p and the degrees of on the unit graph of the integer ring modulo 3p. The result of this research is as follow: GD(G(Z_3p))=█(0;p=2@ 1/24;p=3@ (p+2)/3p(3p-1);p>3) ARABIC: لتكن R حلقة مع العنصر الحيادي بالنسبة إلى الضرب و e∈R عنصر الحيادي بالنسبة إلى الضرب و U(R) مجموعة العناصر القلوبة في R (U(R) = {x∈R|x∙y=e,y∈R}). الرسم البياني القلوب G(R) هو رسم بياني الذي يحتوى العناصر R كقمة الرأس و الرأسان المختلفان, xوy, يتصلان مباشران إذا وفقط إذا x+y∈U(R). مؤشر Gini الدرجات للرسم البياني H هو مجموع فرق أي الدرجتان في الرسم البياني H الذي يقسم بالضرب بين عدد العناصر و معدل الدرجات في الرسم البياني H أو: GD(H)=∑_(u,v∈V(H)@u≠v)▒|deg(u)-deg(v)|/((n^2)E[D^* (H)]) حيث كل من deg(u) و deg(v) هو درجة u و v في الرسم البياني H و n هو عدد العناصر في المجموعة الرؤوس في الرسم البياني H و E[D^* (H)] هو معدل الدرجات في الرسم البياني H. تهدف هذه الدراسة لتحديد صيغة مؤشر Gini الدرجات على الرسم البياني القلوب للحلقة العدد الصحيحة مودولو 3p بp هو عدد الأولي بكيفية تحديد العناصر القلوبة في الحلقة العدد الصحيحة مودولو 3p و تحديد الدرجات في الرسم البياني القلوب للحلقة العدد الصحيحة مودولو 3p . والنتيجة هذه الدراسة هي كالتالي: GD(G(Z_3p))=█(0;p=2@ 1/24;p=3@ (p+2)/3p(3p-1);p>3